On Kulkarni's theorems on degree reduction for polynomial modules

Abstract

In a recent paper [13], Kulkarni proves that, for $i\ges 0$, one has

$\Ext^{i}_{G}(M \otimes \delta(\mu), \nabla(\lambda))\cong \Ext^{i}{G}(M, \nabla(\lambda/\mu))$

– an isomorphism of spaces of extensions of rational modules. Here G is a general linear group scheme, $\lambda, \mu$ are partitions and M is a polynomial G-module. The modules $\Delta (\mu)$, $\nabla (\lambda)$ and $\nabla (\lambda/\mu)$ are respectively the Weyl module labelled by $\mu$, the induced module labelled by $\lambda$ and the skew module labelled by $\lambda /\mu$. A similar result is given in which the roles of $\Delta$ and $\nabla$ are interchanged.

Our purpose here is to set this result in the context of the representation theory of an arbitrary reductive group. Specifically, we prove a simple result which is valid for general reductive groups and derive Kulkarni's results from this. For convenience we work over an algebraically closed field k.